Monday, May 04, 2009

Rheological Math

The math in rheology gets to be really hairy at times, largely because of the constituitive equation. The basic equation is t = m dg/dt, which is deceptively simple. Both the stress, t, and the strain g are 3 x 3 tensors, and m, well m can be just about anything you can imagine as long as your imagination is limited to nightmares. Keep in mind that rheological fluids of interest have a memory of what has just happened to them, a fading memory, but a memory nonetheless, and that this memory needs to be incorporated into the relationship. The only successful efforts are limited to simple flow patterns that allow for simplifications, the creation of linearities and other limitations. In most cases, the equations still can't be solved analytically (in closed form) but need numerical analysis.

Tensors are more than just matrices in the same way that vectors are more than just a row or column of numbers. They are objective quantities, meaning that the relationship between them needs to be valid in all references frames, not just an arbitrary one. A really common example in rheology textbooks is to show that if you study simple pipe flow on a rotating turntable, then with non-objective equations, the viscosity can depend on the rotational speed.

Rumor has it[1] that Einstein, who's first major result was the Stokes-Einstein equation, gave up on the field of rheology as it was too difficult. He went on to something considerably easier - relativity, an area of physics which required the development of new mathematics - objective tensors.

Notes[1] ...which means that I've heard it said, but despite looking everywhere, can't verify it. The facts are true, it's just the motivations that are questionable.